Few theoretical results are known about the convergence of BiCG. For
symmetric positive definite systems the method delivers the same
results as CG, but at twice the cost per iteration. For nonsymmetric
matrices it has been shown that in phases of the process where there
is significant reduction of the norm of the residual, the method is
more or less comparable to full GMRES
(in terms of numbers of
iterations) (see Freund and Nachtigal [102]). In practice
this is often confirmed, but
it is also observed that the convergence behavior may be quite
irregular, and the method may even break
down. The breakdown
situation due to the possible event that
can be circumvented by so-called
look-ahead strategies
(see Parlett, Taylor and Liu [172]). This
leads to
complicated codes and is beyond the scope of this book. The other
breakdown situation, ,
occurs when the
-decomposition fails (see the theory subsection
of §), and can be repaired by
using another decomposition. This is done in the version of QMR
that we adopted
(see §).

Sometimes, breakdown or near-breakdown situations can be
satisfactorily avoided by a restart at the iteration step immediately
before the (near-) breakdown step. Another possibility is to switch to
a more robust (but possibly more expensive) method, like GMRES.